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The largest numbers


geordief
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Are there numbers that are to large for us to represent?

Finite numbers...

If we calculate the number of events/interactions that take place within a defined spacetime volume   is it possible to represent  that in a conventional way ,like 10 to power of some finite number?

Eg  a ball of lead with a mass of 1 kilo.

 

Then ,if we extrapolate and our test volume is increased to include the observed or theorized universe **  that number defies imagination but is there any way to represent it?

Not infinity because it is not ,I think infinite.

Not measurable,quite obviously  but what happens to our number system if we try to give it a number?

 

**ie the number of interactions that might take place over the whole lifetime of a temporaly finite  universe.

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The science of handling large numbers, either large because they have (too) many digits or because the value is too large goes back hundreds of years.

Many schemes for breaking up calculations and other data processing were developed during that time before computers.
But we still have to use some of them today as the data is too large to 'fit' into a single computer.

The instances of this were originally developed by those wanting to produce accurate scientific tables eg Roemer and Kepler, Napier.

Later insurance companies and actuaries carried on this development.

Most recently the largest data handling has been done by meteorological workers.
This started when Walker first started gathering global data via the electric telegraph.

So your idea of event data gathering is not so far from the mark.

Edited by studiot
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I think that for any given alphabet, if both string length and representation time are bounded, then there exists largest representable number. If either one of them is unbounded, then the representable numbers are unbounded as well.

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9 minutes ago, John Cuthber said:

Probably not because we can get creative about the way we express them.
https://en.wikipedia.org/wiki/Knuth's_up-arrow_notation

This constitutes an extension of the alphabet -- re:

11 minutes ago, Genady said:

I think that for any given alphabet, if both string length and representation time are bounded, then there exists largest representable number. If either one of them is unbounded, then the representable numbers are unbounded as well.

 

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I understand that I can write  "two up arrow   up arrow  four is 65536". Which letters didn't you like?
More generally.

The Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters" (a phrase with fifty-seven letters).

https://en.wikipedia.org/wiki/Berry_paradox

 

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2 minutes ago, John Cuthber said:

I understand that I can write  "two up arrow   up arrow  four is 65536". Which letters didn't you like?

Yes, instead of extending the alphabet we can extend the string length boundary. We also can keep alphabet and string length boundary constant and extend the representation time boundary. All of these allows to extend the largest representable number. However, if all three are fixed, I think a largest representable number exists. 

 

7 minutes ago, John Cuthber said:

The Berry paradox 

If we allow for logical paradoxes to be representations, then there is a simpler way to represent any number, e.g., "The number twice as big as the largest number we thought was possible."

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11 minutes ago, John Cuthber said:

We already thought that number was possible.

 

Anyway, logical paradoxes are not representation. There are ways to avoid logical paradoxes, also described in your linked article.

In any case, a number is independent of representation, not defined by it. Different representations can represent the same number, if their string lengths and time limits are large enough.

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You might be interested in this:

https://en.wikipedia.org/wiki/Graham's_number

https://en.wikipedia.org/wiki/Large_numbers

The thing about these large numbers is not just, of course, how big they are. You could always talk about Graham's number +1, and that would be bigger. It's rather about humongously big numbers that somehow are significant in one part or another of mathematics.

Graham's number is really really big in the sense that people seem to be quite uncertain about most of its digits. So in that sense it's very peculiar. Not at all like powers of ten. It's kind of unwieldy in the extreme.

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1 hour ago, joigus said:

You might be interested in this:

https://en.wikipedia.org/wiki/Graham's_number

https://en.wikipedia.org/wiki/Large_numbers

The thing about these large numbers is not just, of course, how big they are. You could always talk about Graham's number +1, and that would be bigger. It's rather about humongously big numbers that somehow are significant in one part or another of mathematics.

Graham's number is really really big in the sense that people seem to be quite uncertain about most of its digits. So in that sense it's very peculiar. Not at all like powers of ten. It's kind of unwieldy in the extreme.

Thanks.Even small numbers present me with difficulty  and so I don't think I will manage to understand how Graham's number is constructed.

 

I suppose numbers can be considered as maps  that need not have a territory. 

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26 minutes ago, geordief said:

Thanks.Even small numbers present me with difficulty  and so I don't think I will manage to understand how Graham's number is constructed.

 

I suppose numbers can be considered as maps  that need not have a territory. 

My pleasure. I don't know how Graham's number is constructed either.

A good principle to organise (integer, counting) numbers by scale (in physics) could be perhaps considering this:

Small numbers: Number of people in a room (somewhere between 10 and 102=100)

Moderately big numbers: Number of atoms in a typical piece of matter (1023)

Big numbers: Number of photons in the universe (1090)

Really big numbers: Permutations of big numbers or number of ways to re-arrange big numbers: (close to NN/eN)

The last one is called Stirling's approximation. It's a way of "taming" really big numbers by avoiding them directly (knowing all their digits) and using instead a ballpark way of dealing with them.

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7 hours ago, geordief said:

Are there numbers that are to large for us to represent?

Finite numbers...

If we calculate the number of events/interactions that take place within a defined spacetime volume   is it possible to represent  that in a conventional way ,like 10 to power of some finite number?

It isn't just a matter of absolute magnitude alone. The largest defined finite integers (such as Graham's number) have a very limited set of prime factors, or are very closely related mathematically to such (eg 10^100 + 1).

One can easily envisage integers that are vanishingly small in comparison to Graham's number, but are far more difficult define uniquely due to the complexity of their prime factor composition and hence their mathematical remoteness from our established notational shortcuts.

The problem then reduces to finding the smallest integer that cannot be uniquely defined within a computable space.

A couple of approaches spring to mind.

Say we set an arbitrary bound on our computer processor to 2^64 bit arithmetic operations and addressing.

Recognising that our decimal based counting system is itself an abbreviation of the numbers, we can now count in base 2^64 up to 2^64 digits. That system tops out at 18,446,744,073,709,551,616^18,446,744,073,709,551,616 (>10^(19*10^19)) 

A second approach may be to recognise that all integers may be represented by Producti=1,n (Pi^yi) where Pi is the ith prime number. Setting the same bit-width bounds on n and y, the first number that fails to compute will be the measly 2^2^64 ~ 10^(8*10^19), but since the majority of referable primes will have magnitudes far in excess of 2^64, the number field will extend far beyond this.

 

 

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1 hour ago, Genady said:

I'm trying to understand this metaphor. Maps of what?

Well numbers can be  a model of real things like the events/interactions in my OP and so  those events are the territory  with the numbers being the maps.

 

Since the numbers can be of any size  , a large enough one   cannot be mapped  onto the real world (the set of real events) and so I thought you could consider them (the sufficiently  large numbers )as a kind of map without a corresponding territory.

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5 minutes ago, geordief said:

Well numbers can be  a model of real things like the events/interactions in my OP and so  those events are the territory  with the numbers being the maps.

 

Since the numbers can be of any size  , a large enough one   cannot be mapped  onto the real world (the set of real events) and so I thought you could consider them (the sufficiently  large numbers )as a kind of map without a corresponding territory.

I think I understand now. And tend to disagree. I think that any number can be mapped onto something in the real world.

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13 minutes ago, Genady said:

I think I understand now. And tend to disagree. I think that any number can be mapped onto something in the real world.

Even if the real world is finite?

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I seem to remember something called Moore's law.

 

Applying this to computer memory I wonder does anyone remember the ZX81, which had 1K of memory

Then the spectrum which had a whopping 16K   so much more space.

 

scale up  few years and we have the laughingly called 1M PC

and onto to today's computers in the gigabyte range.

 

Big is getting bigger and bigger by Moore's law.

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6 minutes ago, geordief said:

So if the number of events in the real world is E  and a number,N >E then you are saying that N can be mapped to E?

 

If there are E events, then there are E2 pairs of events, which represent, e.g., causal relations between the events. E2 > E.

Edited by Genady
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18 minutes ago, Genady said:

I think I understand now. And tend to disagree. I think that any number can be mapped onto something in the real world.

The earth's surface is almost exactly 2x10^84 square Planck units. I vaguely remember reading that something unpleasant happens when you try storing that much information on a limited surface.

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